In 1989, Peebles presented a new method which made it possible to trace galaxy orbits
back in time, provided that they are reasonably laminar, only using the current position
and mass of the galaxies as input, in addition to the cosmological model. This method
has later been known as the *action variational principle* (**AVP** henceforth), and is in practise a variant of the classical
Hamilton's principle from mechanics. The principle has through the years showed its value
in the study of the formation of large-scale structures in the Universe through several
publications by Peebles and others. The method's ability to give the position of each
galaxy at any given time throughout the history of the Universe from a redshift of about
100 until today, has made it possible to, for example, predict the current velocities,
predict current distance given redshift, estimating the primeval mass density
fluctuations, and to give indications of which cosmological model that correctly
describes the Universe.

As one can see, the AVP has a wide range of applications, and could in theory give
exact results if one was able to accurately recreate the orbits of the galaxies. But,
unfortunately, the accuracy of the positions is dependent on quite a few aspects: a
complete and accurate catalogue of galaxies containing celestial coordinates,
distances/redshifts, and the mass; and one need to know the age of the Universe. Such a
catalog does not exist as of today, and the age estimates are at best qualitative. In
addition, one has a problem concerning the dark matter. The AVP assumes that the dark
matter is distributed in the same manner as visual matter--that they both are clumped
together in galaxies. The galaxies are therefore often referred to as ``mass tracers'',
assuming that light trace mass. In other words: if there is a mass component not
visually accessible that are *not* distributed in the same manner as galaxies, it
is ignored by the AVP. All these uncertainties in the input data naturally makes the
output from the AVP equally indefinite.

Yet another problem is the non-linearity when applying the AVP to dense systems of mass tracers, which results in a large number of possible solutions. This can not be avoided in any other way then to simply restrict the AVP to sparse systems of mass tracers, for example by treating a group of galaxies as one particle.

In spite of all these difficulties, the AVP has proven to be surprisingly accurate when applied to a selection of Local Group-galaxies [see e.g.][]P-89, where the observations are of acceptable quality. When applied to a somewhat deeper systems, which for example include galaxies in the immediate neighbourhood of the Local Group, the principle still seems to give a fairly accurate picture. It is when going to distances of a few megaparsec that the inaccuracy becomes most noticeable. Hence, with the observations we have today, the appliance of the AVP is limited to the Local Group and the Local Neighbourhood.

The AVP has similarities to a conventional N-body approach in that it considers the particles as singular points, and that it calculates the orbits of these particles. The major difference between the two methods is the initial conditions. The N-body approach uses six initial conditions (position and velocity in three dimensions) and integrate those forward in time, while the AVP uses a boundary-value approach where the velocities are known at the initial time and the positions are known at the final time. The basic idea of the AVP is to start with a parameterization of the orbits that satisfies these boundary conditions and then adjusts the parameters until a stationary point of the action is found. This makes the AVP considerably more computationally costly than the N-body approach since the former in practise is an optimizing problem in the strongly non-linear regime. All in all, one will never be able to apply the AVP to as large systems as one can with N-body simulations. The reason of pursuing the AVP is that, unlike the N-body calculations, which can only reproduce the current state of a system in a statistical sense, the AVP can fit the current state exactly.

Mon Jul 5 02:59:28 MET DST 1999